Variable automata over infinite alphabets

Abstract Automated reasoning about systems with infinite domains requires an extension of automata, and in particular, finite-word automata, to infinite alphabets . We introduce and study variable finite automata over infinite alphabets (VFAs). VFAs form a natural and simple extension of regular automata, in which the alphabet consists of letters as well as variables that range over the infinite alphabet domain. Thus, VFAs have the same structure as finite automata, except that some of the transitions are labeled by variables. We compare VFAs with existing formalisms, and study their closure properties and classical decision problems. We further identify and study the deterministic fragment of VFAs (DVFAs). We show that while DVFAs are sufficiently strong to express many interesting properties, they are closed under the Boolean operations, and their nonemptiness and containment problems are decidable. We describe a determinization process for a determinizable subset of VFAs. Moreover, we show that DVFAs have a canonical form, making them a particularly robust model that is easy to reason about and work with. Building on these results, we construct an efficient active learning algorithm for DVFAs, based on the $$L^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> learning algorithm for regular languages.